Solve each equation in Exercises 65–74 using the quadratic formula. x² + 5x + 3 = 0

Verified step by step guidance

1. Identify the coefficients in the quadratic equation. The general form of a quadratic equation is ax^2 + bx + c = 0. Here, a = 1, b = 5, and c = 3.
2. Recall the quadratic formula: $x=\frac{-b\pm \sqrt{b^2-\; 4ac}}{2a}$. This formula is used to solve any quadratic equation.
3. Substitute the values of a, b, and c into the quadratic formula. Replace a with 1, b with 5, and c with 3: $x=\frac{-5\pm \sqrt{5^2-\; 4\left(1\right)\left(3\right)}}{2\left(1\right)}$.
4. Simplify the discriminant (the part under the square root): $5^2-\; 4\left(1\right)\left(3\right)$. Calculate $5^2$ (which is 25) and subtract $4\left(1\right)\left(3\right)$ (which is 12).
5. Write the simplified quadratic formula with the discriminant value substituted. If the discriminant is positive, proceed to calculate the two possible solutions for x using the ± symbol. If the discriminant is zero, there will be one solution. If the discriminant is negative, the solutions will involve imaginary numbers.

Key Concepts

Quadratic Equation

A quadratic equation is a polynomial equation of the form ax^2 + bx + c = 0, where a, b, and c are constants, and a ≠ 0. The solutions to this equation can be found using various methods, including factoring, completing the square, or applying the quadratic formula.

Quadratic Formula

The quadratic formula is a mathematical formula used to find the roots of a quadratic equation. It is expressed as x = (-b ± √(b² - 4ac)) / (2a), where a, b, and c are the coefficients from the quadratic equation. This formula provides a systematic way to calculate the solutions, even when the equation cannot be easily factored.

Discriminant

The discriminant is the part of the quadratic formula under the square root, given by b² - 4ac. It determines the nature of the roots of the quadratic equation: if the discriminant is positive, there are two distinct real roots; if it is zero, there is one real root (a repeated root); and if it is negative, there are two complex roots. Understanding the discriminant helps in predicting the type of solutions without solving the equation.